Page 51 - Calculus with Complex Numbers

4.7 M ethod 3: U sing the residue theorem
Theorem 3 above says that if flz) has a primitive on the closed contour y, then

Theorem 4 (Cauchy's theorem) lf y is a closed contour and if flz) has a
derivative on y and evelywhere irlside y, then

f U) dz = 0.
r
Proof See Appendix 1.

We can use Cauchy's theorem to show (for a third time) that if y is the unit
circle then

zn dz = 0
r
for n k: 0. For n < 0 Cauchy's theorem does not tell us anything, since zn then
has a singularity at z = 0 which is inside y .
Cauchy's theorem might appear at lirst sight to be rather trivial. However it
turns out to have far reaching consequences as we shall shortly see.

Corollal'y 1 lf the contours n , yz have the same end points a b and if f (z) is
differentiable on n , yz and between them, then (Figure 4.6)

flz) dz = flz) dz.
/1 /2
Proof lf y = yz - n , then we can apply Cauchy's theorem to y to obtain

0 = J flz) dz = J ./'U) dz - J ./'(Z) dZ'
J / J /% J /1

Ffgure 4.8